Curved Space

A Variant of ST3D Chess
by L. Lynn Smith

This variant is rather complex to initially visualize but its reasoning will become
apparent with familiarity. The Attack Platforms create a "worm-hole" across the field,
and "fold space" on those Fixed Levels which they are adjacent. Although these rules
utilize only four Attack Platforms, it is possible to use more. But that way leads to
madness.

Equipment

1 ST3D playing field with four Attack Platforms

1 Standard set of Chessmen

Mark each of the Attack Platforms with a number between 1 and 4. This will be used
to denote their order in the "worm-hole"

Rules

Play begins with all pieces off the field, including the Attack Platforms. Each player
in turns introduces an Attack Platform to any point on the playing field. It is not
necessary that they be placed in their numbered order.

After all four Attack Platforms have been placed, each player in turn places their King
to their particular Fixed Level. Then each may either, introduce a new piece or move a
piece on the field.

All pieces are introduced by the player to their particular Fixed Level.

Piece move upon the Fixed Levels accordingly:

PAWNs step orthogonal without capture or step diagonal with capture.

KNIGHTs leap one orthogonal then one forward diagonal.

BISHOPs slide diagonal.

ROOKs slide orthogonal.

QUEEN slides orthogonal or diagonal.

KING steps orthogonal or diagonal.

Movement of Pieces from Fixed Levels to Attack Platforms and Beyond:

This is where the concept of "worm-holes" and "folded space" is used. First lets look at
the 4x4 area of the Fixed Level.

4 [ ][ ][ ][ ]
3 [ ][w][x][ ]
2 [ ][y][z][ ]
1 [ ][ ][ ][ ]
a b c d

The four central cells of each Fixed Level and the four cells of each Attack Platform will
make up the structure of the "worm-hole" which will run through the field according to
specific conditions.

The four central cells have been noted with the letters "w", "x", "y" and "z". These will
help explain their connection to the Attack Platforms and other Fixed Levels.

Attack Platforms become connected to these central cells when they are placed on one of the
points adjacent to that particular Fixed Level. The relation of the cells of the Attack
Platforms is thus:

2 [w][x]
1 [y][z]
a b

Orthogonal "worm-hole" movement would be considered a move from a "w" cell to the next
"w" cell. It would follow that that "x" is orthogonal to "x", etc.

Diagonal "worm-hole" movement would be consider a move from a "w" cell to the next "x"
or "y" cell. It follows that "x" is diagonal to "w" or "z", etc.

Triagonal "worm-hole" movement would be consider a move form a "w" cell to the next "z"
cell. It follows that "y" is triagonal to "x".

If there are no Platforms on any of the points of a Fixed Level, its four center cells
are connected to the four center cells of the next Fixed Level. In this order, White
to Neutral to Black. Or, Black to Neutral to White. Attack Platforms modifies these
connections.

With an Attack Platforms on the points of Fixed Levels, the center four cells of that
particular Fixed Level are now oriented to these particular Platforms according to their
numbered order and orientation.

A Platform in the up-right position is considered "above" its particular Fixed Level.
A Platform in the inverted position is considered "below" its particular Fixed Level.

Confusing!!? Let's have an example. A playing field begins with Attack Platforms at
Wa1u, Nd1d, Bd4u and Na4u. These are numbered respectively "1" through "4". Therefor
the structure of the "worm-hole" can seen as starting at the four central cells of the
White Fixed Level. So its "upward" structure could be expressed as White-Wa1u-Nd1d-
Neutral-Black-Bd4u-Neutral-Na4u. It's downward structure would then be the reverse.

Whew!!! Let's look at that again. But this time the "1" Attack Platform at Wa1u is now
at Ba1d. So the "upward" order is Ba1d-Black-Nd1d-Neutral-Black-Bd4u-Neutral-Na4u. The
connection between the White Fixed Level and the Neutral Fixed Level is a possible "branch"
at each instant of the appearance of the Neutral position in this structure. So a piece
can exit the "worm-hole" to the White Fixed Level on either appearance at the Neutral Fixed
Level. And can enter the "worm-hole" from the four center cells of the White Fixed Level
at either appearance of the Neutral Fixed Level in the "worm-hole" structure.

Still confused? Well, let's not dwell on this. Let's now look at piece movement through
the "worm-hole".

Pieces are allowed to perform their above mentioned moves across the four central cells of
a particular Fixed Level. But while moving through the "worm-hole" they are subject to
certain restrictions and new rules.

All the following are "worm-hole" moves:

PAWNs step orthogonal without capture or step diagonal with capture.

KNIGHTs steps triagonal.

BISHOPs slide diagonal.

ROOKs slide orthogonal.

QUEEN slides orthogonal or diagonal.

KING steps orthogonal or diagonal.

There is a unique feature to the diagonal "worm-hole" slide. This can be either clockwise
or counter-clockwise. Clockwise would be movement from the "w" cell to the "x" cell to the
"z" cell to the "y" cell. Counter-clockwise would be movement from the "w" cell to the "y"
cell to the "z" cell to the "x" cell. Players may opt for either direction when beginning
a diagonal slide, and will maintain this direction throughout the slide. This can be verified
by touching each appropriate cell along the way.

Captures can be preformed within the "worm-hole".

Movement of Attack Platforms:

Here the players have several options:

Forbid Attack Platform movement. Once they are placed, they do not move throughout the
game.

Allow a player to move an empty Platform to any vacant point or orientation on the field
as an additional move upon the preforming of a capture.

The game is won by check-mating the opponent King.

For both clarity, and sanity's sake, players might maintain Platforms to specific points
on each Fixed Level, especially in four Attack Platform games. Such as the "1" Attack
Platform will always be placed on the a1 point of any Fixed Level, the "2" Attack Platform
will always be place on the a4 point of any Fixed Level, etc. This will aid greatly in
being able to quickly evaluate the nature of the "worm-hole". And in a four Platform
game it would then be un-necessary to number the Platforms, since their location will
now determine their order.

AUTHOR'S AFTERWORD

Allow me to express my sincere apologies to those who now suffer from bouts of confusion
and\or frustration. This game is not for the weak of heart or mind. The structure of the
"worm-hole" can be a confusing thing, and can take a large variety of shapes from game
to game. By now, the reader might have realized that this structure can really become
strange and convoluted. Well, that's the nature of Curved Space and it's completely
un-forgiving.
Written by L. Lynn Smith.
Webpage created: October 30, 2003. Last modified: February 4, 2004.
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